No, it does not. I and others have been supporting this script for years and would have found such a thing by now. And, besides, there is only *1* developer of the script (cRYOa). There are 3 who still maintain it and offer unofficial patches to it and answer questions on the forum and in the Invision channel listed at the bottom of this post (XMog, L|B, and myself).
The only way there would be a keylogger would be if someone modified the script. I would warn you that there are versions of Invision (and other scripts) out there that have been modified to include things that you probably don't want. Always download scripts from approved locations and be wary of any you don't trust.
The official place to get Invision from is
http://www.i-n-v-i-s-i-o-n.com/ and the forum there tells you where you can get the latest patch.